What is the kernel of $K[x^2,x^3][T] \to K[x]$, defined by $T \mapsto x$?

Consider $K[x^2,x^3] \subset K[x]$, where $x$ is an indeterminate over a (zero characteristic) field $K$.

Clearly, $x$ vanishes the following polynomials $\in K[x^2,x^3][T]$:

$f(T)=x^2T-x^3$, $g(T)=T^2-x^2$, $h(T)=T^3-x^3$, etc.

I wish to find the kernel of the surjective homomorphism $K[x^2,x^3][T] \to K[x]$, defined by $T \mapsto x$; in other words, what are the generators of the kernel? (Obviously, $f(T), g(T), h(T)$ are all in the kernel).

Since $T^2-x^2$ is monic, every element in $K[x^2,x^3][T]$ can be written as $(T^2-x^2)Q(T)+AT+B$, where $A,B\in K[x^2,x^3]$ and $Q(T)\in K[x^2,x^3][T]$. Such an element will be in the kernel if and only if $AT+B$ is in the kernel, which means $Ax+B=0$, or $B=-Ax$.

Suppose it is and write $A=a_0+a_2x^2+a_3x^3+\dots+a_nx^n$; then $-B=a_0x+a_2x^3+\dots+a_nx^{n+1}$ and, since $B\in K[x^2,x^3]$, it must be $a_0=0$.

Thus $A=x^2(cx+f(x))$, for some $f\in K[x^2,x^3]$ and $c\in K$, so $$ AT+B=x^2(cx+f(x))T-x^3(cx+f(x))=c(x^3T-x^4)+f(x)(x^2T-x^3) $$

Thus an element in the kernel belongs to the ideal generated by $T^2-x^2$, $x^2T-x^3$ and $x^3T-x^4$.

The converse is clear.

For the problem of division, let's look at it in a more general setting. Let $R$ be a commutative ring and let $f(T)$ be a monic polynomial in $R[T]$. Then, for every polynomial $g(T)\in R[T]$ there exist $q(T)$ and $r(T)$ such that

1. $g(T)=f(T)q(T)+r(T)$
2. $\deg r<\deg f$ (where, as usual, the degree of the zero polynomial is $-\infty$).

The proof is the same as in the field case, by induction on the degree of $g$. If $g=0$ there is nothing to prove, so assume $g\ne0$, $g=a_mT^m+a_{m-1}T^{m-1}+\dots+a_0$.

Write $f(T)=T^n+f_0(T)$, with $\deg f_0<n=\deg f$ (it's not restrictive to assume $n>0$, or the result is obvious). If $\deg g<n$ we take $q=0$ and $r=g$. Otherwise we consider $$ g_0(T)=g(T)-a_mT^{m-n}f(T) $$ which has degree less than $m$. By the induction hypothesis, $$ g_0(T)=f(T)q_0(T)+r(T) $$ with $\deg r<\deg f$. Then $$ g(T)=g_0(T)+a_mT^{m-n}f(T)= f(T)(a_mT^{m-n}+q_0(T))+r(T) $$ and we are done.

It's easy to extend the result to polynomials $f$ having invertible leading coefficients, by considering $u^{-1}f$, where $u$ is the leading coefficient: then $g(T)=u^{-1}f(T)q(T)+r(T)$ and so $u^{-1}q$ is the quotient and the remainder is the same.

Let's see uniqueness. Suppose $$ g(T)=f(T)q_1(T)+r_1(T)=f(T)q_2(T)+r_2(T) $$ with $\deg r_1<\deg f$ and $\deg r_2<\deg f$. Then $$ f(T)(q_1(T)-q_2(T))=r_2(T)-r_1(T) $$ If $q_1-q_2\ne0$, then the degree of $f(q_1-q_2)$ is at least equal to the degree of $f$ (because $f$ is monic), which is a contradiction, because the degree of $r_2-r_2$ is less than $\deg f$.

In your setting, $R=K[x^2,x^3]$ and $f(T)=T^2-x^2$.